TRANSVERSE VIBRATION OF A GEAR WHEEL

Transcription

1 ISSN TRANSVERSE VIBRATION OF A GEAR WHEEL Stanislaw Noga, Rzeszow University of Technology, ul. W. Pola, Rzeszow, Poland. Abstract: In the paper, transversal vibration of the annular plate related to the toothed gear is studied using finite element representation. The plate is fixed in the hollow shaft. The base model contains all essential construction of the aviation engine toothed gear. The algorithm to identify the proper distorted mode shapes is presented. The pre stress effect is included using the FE code system standard procedure. It is important to note that the data presented in the paper is brought the practical advice to design engineers. Key words: annular plate, transversal vibration, natural frequencies, mode shapes, toothed gear. 1. INTRODUCTION Development of modern engineering requires technical equipment which is characterized by high durability and operational reliability. This particularly applies to the aviation industry components and assemblies. Each newly produced assembly must pass a cycle of statical and dynamical computations which allow it to be used in further experimental tests. One of the essential parts of the aviation engines is the toothed gear. It results from the fact that these elements may produce the vibration of aviation engine subassembly during work. Because of this reason, it is beneficial to conduct numerical computations at the design stage, which could limit the consequences of the gear vibration. Dynamical analysis of that systems is mainly focused on the free vibration analysis to determine the natural frequencies and natural mode shapes of vibration [3]. The most popular in the engineering applications are disk gears. From the vibration theory point of view such gear may be treated as the annular plate mounted on the hollow shaft [6]. The problem of transverse vibration of annular plate systems is a well known problem in structural dynamics. Fundamental theory of plate system is developed in [6]. Transversal vibrations of annular plates are studied in works [1,, 4, 5]. In the paper [4] solution for the plate with linear variable thickness is presented. In the work [5] the boundary problem is solved for the plate with nonlinear variable thickness. In both works the Ritz method is utilized. In the article [1] transversal vibrations of a plate with radial through cracks are studied using finite element method. Transversal vibrations of annular composite plate are analyzed in work [] using finite element representations. Paper [7] presents introductory studies that deal with the vibrations of the gear wheel. 137

2 ISSN In this article free transverse vibrations of the toothed gear are analyzed using finite element technique. The gear taking into consideration is treated as the annular plate with the complex geometry, mounted on the hollow shaft. The plate has the geometrical discontinuity. For the system under investigation the finite element model is elaborated. In the work results of numerical solution for this system are displayed. The algorithm to identify the proper distorted mode shapes is presented. This work continues the recent author s investigations concerning the dynamics of structures [8].. FORMULATION OF THE PROBLEM The mechanical model of the toothed gear taking into consideration, consists of the hollow shaft and the annular plate with the complex geometry. Geometrical model of the system is presented in the Fig. 1a. Such structure is coon used in the aviation gearboxes. a) b) h p R a d z dw1 d k d 0 h z θ r d w w d p Fig. 1. (a) Model of the system, (b) Annular plate model In this gear model the toothed ring is omitted. It is assumed that the outer diameter of the gear d z is equal the pitch diameter of the gear. The remainder dimensions of the gear are defined as per the Fig. 1a. The annular plate of the gear has the discontinuity as it is shown in the Fig 1a. According to the plate theory, the partial differential equations of motion for the free transversal vibrations may be written in the form [6] 3 Eh 11 r 1 1 r r r w 0 w t where w = w(r, θ, t) is the transverse plate displacement (see Fig. 1b); r, θ, t are the polar coordinates and the time; h, R, a are the plate dimensions; E, ν are the Young s modulus of elasticity and the Poisson s ratio; ρ is the mass density and γ = ρh. Because it is quite difficult to use the model based on partial differential equation for the analysis of plates that have the geometrical discontinuity, the finite element (FE) representation is employed to solve the free vibration problem. Introducing the finite element assumption, leads to the equations of motion of the system that may be written in the form [3] Mu Ku 0 ( ) where M and K are, the global mass and global stiffness matrices (made up by proper assembly of the element matrices); ü and u are the nodal acceleration and nodal displacement ( 1 ) 138

3 ISSN vector, respectively. The global mass and stiffness matrices are assembled from the element matrices that are given by [3] e e T e M N N dv, e V q q e V e e T K B E 0 BdV ( 3 ) where ρ (e) is the mass density coefficient for an element e; V (e) is the volume of the element e; N q is the matrix of the element shape functions; B and E 0 are, the element shape function derivatives, and the elasticity matrices, respectively. The natural frequencies of the system may be obtained by solving the eigenvalue problem K Mu 0 ( 4 ) where ω is the natural frequency and ū is the corresponding mode shape vector which can be obtained from equation (4). The number of eigenpairs (ω i, ū i ) corresponds the number of degree of freedom of the system. The block Lanczos method is employed to solve the eigenvalue problem (4) [9]. 3. NUMERICAL ANALYSIS Numerical analysis results of the toothed gear taking into consideration free vibration are obtained using the finite element representation. The natural frequencies and natural mode shapes are established. The parameters characterizing the system used in calculation are shown in table 1. d z d p d k d w1 d w d 0 table 1. Parameters characterizing the system h z h p E MPa kg/m ,7 6,7 18,8,4 7,3 0,3,08*10 5 7,83*10 3 The model consists of the complex geometry divided into finite elements. The ten node tetrahedron element with three degree of freedom in each node is used to solve the problem. The prepared FE model is shown in Fig.. The complete description of the FE model and the boundary conditions may be found in the paper [7]. Fig.. The FE model of the system Circular syetry of the plate is the reason of, each solution related to the modes consist of the nodal diameters has two identical mode shapes and differ from one another only by an angular rotation of α = π/(n), where n determines the number of nodal diameters. Because the system under investigation is more complicated and has geometrical 139

4 ISSN discontinuity, some mode shapes related to the transversal vibrations of the gear plate are distorted. It introduces the problems especially for aviation engineers to identify the proper solutions. The following procedure allows to solve this problem. At first two auxilary models of the annular plate are prepared (see Fig. 3). Basic dimensions of the models are related to the model of the toothed gear. The second model includes the rim. Then for such models the dynamical analysis is executed to achieve the natural frequencies and natural mode shapes. d 01 d 0 Fig. 3. Model 1 and model During analysis, dimensions of the port diameters (d 01 and d 0 ) for both models are changed according with the table. After that achieved results are compared with each other. The qualitative and quantitative difference are taken into account. table. Hole diameters of auxiliary models model 1 i model d 01, d 0 [] Proposed approach allows to see the distortion of the toothed gear under investigation mode shapes because of the port. At the end results from the model 1 are compared with results from the model shown in Fig. (so called base model). The second model is employed when the solution from the first model is not satisfied. For presented models the computations are conducted until the natural frequency connected with the mode shape related to the eight nodal diameters is achieved. The calculations are executed with making an assumption that systems are rotated with the angular velocity equal φ = 1047 [rad/s]. The gyrate effect is included by using the pre stress effect standard procedure [9]. The results of the calculation are presented below. Due to space limitation only two examples are presented in this paper. For the base model (see Fig. ) both solutions related to the modes consist of the nodal diameters are presented and for the remainder models only one solution is displayed. Values of the natural frequencies related to presented mode shapes are shown in table 3. Fig. 4. The mode shapes corresponding to frequency ω 5 (base model) 140

5 ISSN Fig. 5. The mode shapes corresponding to frequency ω 5 (model ) This is special case where the solution contains two mode shapes that have different forms and different values of the corresponding natural frequencies. It was easy to identify the first mode shape. To identify the second mode shape the model is utilized. For this mode the considerable distortion of the nodal circle and nodal diameters is visible. table 3. Values of the natural frequencies model d 0 [] base model [Hz] model 1 d 01 [] base model [Hz] Fig. 6. The mode shapes corresponding to frequency ω 6 (base model) Fig. 7. The mode shapes corresponding to frequency ω 6 (model 1) In this case the model 1 is utilized to identify the proper solution. Mainly the considerable distortion of the nodal diameters is observed. 141

6 ISSN CONCLUSION The design of modern aviation engine rotating components require the use of advanced numerical software connected with the finite element method. This allows to take into consideration the complex geometry of the analyzed system. This paper deals with the free transverse vibrations of the system related to a toothed gear with the complex geometry. The free vibrations are determined by using the finite element method. The algorithm to identify the proper distorted mode shapes is presented. Apart from the mentioned publications in this article, in available literature, I have not found examples of work related to the problems analyzed in this paper. ACKNOWLEDGEMENT The numerical calculations are supported by the grant from The State Coittee for Scientific Research (KBN) no. MNiSW/IBM_BC_HS1/Przesz./001/ REFERENCES [1] Demir A., Mermertas V., A study of annular plates with radial through cracks by means of sector type element, Journal of Sound and Vibration, 300, 007, [] Efraim E., Eisenberger M., Exact vibration analysis of variable thickness thick annular isotripic and FGM plates, Journal of Sound and Vibration, 99, 007, [3] Ginsberg J. H., Mechanical and structural vibrations, Wiley, New York 001. [4] Kang J. H., Leissa A. W., Three dimensional vibrations of thick, linearly tapered, annular plates, Journal of Sound and Vibration, 17, 1998, [5] Kang J. H., Three dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation, Computers & Structures, 81, 003, [6] Leissa A. W., Vibration of Plates, NASA SP 160; US Government Printing Office, 1969 (reprinted by The Acoustical Society of America, 1993). [7] Noga S., Analysis of toothed gear transverse vibrations, Cracow University of Technology Publishers, Technical Journal, Mechanics, 9 M (008) (in Polish). [8] Noga S., Free transverse vibration analysis of an annular membrane, Vibrations in Physical Systems vol. XXIII (008) [9] Paško, J. - Pavlenko, S. - Haľko, J. Conditions for toothing in two-stage multi-output transmissions. In: Scientific Bulletin : Fascicle: Mechanics, Tribology, Machine Manufacturing Technology. vol. 0, serie c (006), p ISSN [10] Rakowski G., Kacprzyk Z., Finite element method in structural mechanics, Warsaw University of Technology Publishers, Warsaw 005 (in Polish). 14